§7 · Agent 7 · Adversarial audit · 2026-10-17T03:21:52Z

Audit 7.16 · Audit: the comparability objection is correct — three fields, two bridges, and what each bridge needs

proved audit: error

Error. The assessment of Objection 1.8 and both bridges are correct (the spreading-out bridge works label by label, because the Hecke combinations used — dominant lifts, the cancellation of Proposition 1.4, central translates — have a q-independent shape). False: 'Chen–Yi §8.2 needs only T₀ split, which the SO₂-model also satisfies (S = T diagonal)'. In Chen–Yi, (B₀, T₀) is a θ-stable Borel pair, and §8.2 assumes G, B₀, T₀, θ defined over 𝔽_q with T₀ split. For θ(g) = (gᵀ)⁻¹ = Ad(J)(g), the θ-stable Borel subgroups are the stabilisers of the eigenlines of J, spanned by (1, ±i), and the only θ-stable maximal torus inside them is SO₂; they are defined over 𝔽_q, and SO₂ is split, only when q ≡ 1 mod 4. The diagonal torus is the θ-split torus S of Chen–Yi's other (θ-split) Borel pair. The stated conclusion — Lang's count applies to both forms — is true for an independent reason (x_m rational, π₀Stab(x_m) = {±I} with trivial Frobenius action). The conclusion still holds by another argument.
Corrected by: Correction 10.5 · The \(\SO_2\)-model of \(\SL_2/\SO_2\) over \(\F_q\) and Chen–Yi's splitness hypothesis, Correction (repair of A7-16) X2.5 · Chen–Yi §8.2 holds for the \(\mathrm{SO}_2\)-form iff \(q\equiv1\bmod4\); Lang’s count holds in both.

Confirms that a divergence read off the pages' table would compare Betti sheaves over \(\C\) with \(p\)-adic functions under Sakellaridis's hypotheses, and spells out the two bridges needed to compare them. Its side remark that the \(\SO_2\)-model satisfies Chen–Yi's splitness hypothesis is false (see the audit).

Audit 7.16.

Objection 1.8 is correct: the sheaf statement is Betti over \(\C\), Theorem E is \(p\)-adic (characteristic \(0\) in (A1)(A1) Field and group: F is a local field of characteristic 0 (p-adic), and G is split over the integers, with connected fibres and simply connected derived group. Its characteristic-p counterpart is (A1ₚ).), and the trace dictionary links \(\F_q(\!(t)\!)\) with ℓ-adic sheaves over \(\overline{\F}_q\). Calling a Table-1 cell a divergence requires two bridges:

  1. Sheaf bridge \(\overline{\F}_p\leftrightarrow\C\): standard, by generic base change for \(p\) outside a finite set; it suffices for the rows where the function inputs hold for every \(q\). The inertia argument of Theorem 2.4 needs no bridge.
  2. Function bridge \(p\)-adic \(\leftrightarrow\F_q(\!(t)\!)\): needed only for the rows "(SG) under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies."; it requires Sakellaridis's Thm. 8.0.2 in characteristic \(p\), or direct lattice counts.

Moreover, Chen–Yi's §8.2 needs only \(T_0\) split, which the \(\SO_2\)-model also satisfies (\(S=T\) diagonal), so Lang's theorem applies to both \(\F_q\)-forms of \(\SL_2/\SO_2\).

  1. Sheaf bridge. The statement "\(\IC_\mu\) is a summand of \(V_\lambda\ast\IC_0\)" lives on a proper map of finite-type varieties defined over \(\Z[1/2]\). Generic base change (Beilinson–Bernstein–Deligne, §6.1) gives equal decomposition multiplicities over \(\C\) and over \(\overline{\F}_p\) for all \(p\) outside a finite set. Since the function-level inputs used (Theorem 3.9, Proposition 1.3, Proposition 1.4, Lemma 4.8) hold for every \(q\), one may choose \(p\) large; this bridge is standard though not written out on the pages, and it suffices for the \(\GL_4/\Sp_4\) verdict.
  2. Function bridge. Only needed for the rows "(SG) under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies.": \(\GL_{2m}/\Sp_{2m}\), \(m\ge3\), and the failure for \(E_6/F_4\) (\(\GL_n/(\GL_p\times\GL_q)\) is now covered over \(\F_q(\!(t)\!)\) by Lemma 4.8). It needs Sakellaridis's Thm. 8.0.2 in characteristic \(p\) — whose proof uses his Axioms 2.4.1–2.4.2 and Casselman-type computations, not asserted there — or direct lattice counts on the scale of Theorem 4.9. The computations of the restriction map \(r\) in Proposition F and in Theorem 6.6 do not depend on the field, so only the theorem itself is missing.
  3. Consequence for the verdicts. Each of the team's conclusions is proved in one consistent setting: Betti for \(\SL_2/\SO_2\); \(\F_q(\!(t)\!)\) against \(\overline{\F}_q\) for \(\GL_4/\Sp_4\), the transfer theorem and the conditions of Proposition III (with the function half of \(\GL_4/\Sp_4\) over every \(F\)); \(p\)-adic for \(E_6/F_4\). The point of Objection 1.8 that Theorem 1.6 and Proposition 1.7 are the only literally comparable results is right, with Proposition 3.10 and Theorem 4.6 (the same theorem) and Theorem 2.4 (Betti, stronger) added.
  4. Models of \(\SL_2/\SO_2\). The split-model remark of Objection 1.8 is correct; but note that Chen–Yi's §8.2 needs only \(T_0\) split, which the \(\SO_2\)-model also satisfies (\(S=T\) diagonal), so Lang's theorem applies to both \(\F_q\)-forms.

Builds on

Built on by

  • Summary 7.19 · First audit verdict: conclusions (i)–(v) survive with their hypotheses; the only candidate divergence is \(E_6/F_4\)
  • Correction 10.5 · The \(\SO_2\)-model of \(\SL_2/\SO_2\) over \(\F_q\) and Chen–Yi's splitness hypothesis
  • Correction (repair of A7-16) X2.5 · Chen–Yi §8.2 holds for the \(\mathrm{SO}_2\)-form iff \(q\equiv1\bmod4\); Lang’s count holds in both
  • Audit X2.7 · No \(\mathrm{SL}_2/\mathrm{SO}_2\) conclusion of the run rests on the false §8.2 remark
Original record, as the agent wrote it at 03:21 UTC

A1-8 comparability objection is correct: three fields, two bridges, and what each bridge needs

Assessment: correct. The sheaf statement is Betti over \(\mathbb C\); Theorem E is \(p\)-adic (char 0 in (A1)); the trace dictionary links \(\mathbb F_q(\!(t)\!)\) to \(\ell\)-adic sheaves over \(\overline{\mathbb F}_q\). A Table-1 cell pairing a Betti verdict with a '(SG) under (A)' verdict compares two statements with no map between them; calling it a divergence requires two bridges:

  1. Sheaf bridge \(\overline{\mathbb F}_p\leftrightarrow\mathbb C\) (D5 / open item 1). The statement '\(\mathrm{IC}_\mu\) is a summand of \(V_\lambda\ast\mathrm{IC}_0\)' lives on a proper map of finite-type varieties defined over \(\mathbb Z[1/2]\); BBD §6.1 generic base change gives equality of decomposition multiplicities between \(\mathbb C\) and \(\overline{\mathbb F}_p\) for all \(p\) outside a finite set. Since the function-level inputs used (A3-9, A1-3, A1-4, A4-8) hold for every \(q\), one may choose \(p\) large: this bridge is standard though not written out on the pages, and it suffices for (ii). The inertia argument A2-4 needs no bridge at all.
  2. Function bridge \(p\)-adic \(\leftrightarrow\mathbb F_q(\!(t)\!)\). Only needed for the '(SG) under (A)' rows (\(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(m\ge3\); \(E_6/F_4\) failure; \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) is now covered over \(\mathbb F_q(\!(t)\!)\) by A4-8). Needs Sakellaridis Thm. 8.0.2 in characteristic \(p\) (its proof uses Axioms 2.4.1–2.4.2 and Casselman-type computations; not asserted), or direct lattice counts (A4-10 scale). Prop. F's and A6-6's computations of \(r\) are field-independent, so only the theorem itself is missing.

Consequence for the team verdict. Statements (i)–(iv) are each proved in one consistent setting (Betti for (i); \(\mathbb F_q(\!(t)\!)\)/\(\overline{\mathbb F}_q\) for (ii)–(iv), with (ii)'s function half over every \(F\)); (v) is \(p\)-adic. A1-8's point that A1-6 and A1-7 are the only literally comparable results is right, with A3-10/A4-6 (same theorem) and A2-4 (Betti, stronger) added. The split-model remark for \(\mathrm{SL}_2/\mathrm{SO}_2\) is correct but note Chen–Yi §8.2 needs only \(T_0\) split, which the \(\mathrm{SO}_2\)-model also satisfies (\(S=T\) diagonal), so Lang (A4-2) applies to both \(\mathbb F_q\)-forms.

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